Algebra

# Absolute Value Problem Solving

Let $x=70$ be one root of the equation $2\lvert x-10\rvert=x+a,$ where $a$ is a constant. Then what is the other root?

Let $S$ be the solution set of the equation $\lvert 2x-1 \rvert + 3\lvert x-29\rvert=5x-88.$ Then what is $S?$

How many ordered pairs $(x, y)$ are there such that $x$ is a nonzero integer that satisfies $-8\leq x \leq 8$, $y$ is a nonzero integer that satisfies $-2\leq y \leq 2$, and $\lvert x+y \rvert = \lvert x \rvert + \lvert y \rvert?$

Details and assumptions

For an ordered pair of integers $(a,b)$, the order of the integers matter. The ordered pair $(1, 2)$ is different from the ordered pair $(2,1)$.

How many ordered triples of integers $(a, b, c)$ subject to $-20 < a < b < c < 20,$ are there, such that

$\left| a + b + c \right | = |a| + |b| + |c|?$

If $20 < a \leq 21$, what is the value of $\bigg| 2-a- \big| 6- \lvert a-20 \rvert \big| \bigg| ?$

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